Franz E. Schuster

Crofton measures and Minkowski valuations

A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations.

Convolutions and multiplier transformations of convex bodies

Rotation intertwining maps from the set of convex bodies in into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator

Rotation equivariant Minkowski valuations

The projection body operator \Pi, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that \Pi\ maps the set of polytopes in Rn into itself. We show that \Pi\ is the only non-trivial operator with these properties.

Volume Inequalities and Additive Maps of Convex Bodies

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies.

An arithmetic proof of John's ellipsoid theorem

Abstract.Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John’s characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the ‘hard part’ of a generalization of John’s theorem and shows the difficulties of the corresponding ‘easy part’.

Even Minkowski valuations

A new integral representation of smooth translation invariant and rotation equivariant even Minkowski valuations is established. Explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived. Moreover, the action of Alesker’s Hard Lefschetz operators on these Minkowski valuations is explored in detail.

A CHARACTERIZATION OF BLASCHKE ADDITION

A characterization of Blaschke addition as a map between origin-symmetric con- vex bodies is established. This results from a new characterization of Minkowski addition as a map between origin-symmetric zonoids, combined with the use of L evy-Prokhorov metrics. A full set of examples is provided that show the results are in a sense the best possible.

Rotation Invariant Minkowski Classes of Convex Bodies

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1; T2 such that M + T1 = T2, and T1; T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K, which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof.

A Hadwiger-Type Theorem for General Tensor Valuations

Hadwiger’s characterization of continuous rigid motion invariant real valued valuations has been the starting point for many important developments in valuation theory. In this chapter, the decomposition of the space of continuous and translation invariant valuations into a sum of SO(n) irreducible subspaces, derived by S. Alesker, A. Bernig and the author, is discussed. It is also explained how this result can be reformulated in terms of a Hadwiger-type theorem for translation invariant and SO(n) equivariant valuations with values in an arbitrary finite dimensional SO(n) module. In particular, this includes valuations with values in general tensor spaces. The proofs of these results will be outlined modulo a couple of basic facts from representation theory. In the final part, we survey a number of special cases and applications of the main results in different contexts of convex and integral geometry.